Answer
See the explanation below.
Work Step by Step
The given expression on the left side $2\sin {}^{3}\theta \cos \theta +2\sin \theta {{\cos }^{3}}\theta $ can be further simplified as:
$2\sin {}^{3}\theta \cos \theta +2\sin \theta {{\cos }^{3}}\theta =2\sin \theta \cos \theta \left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)$
We have the Pythagorean identity, ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Now, applying the identity, the expression can be simplified as:
$\begin{align}
& 2\sin \theta \cos \theta \left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)=2\sin \theta \cos \theta .\left( 1 \right) \\
& =2\sin \theta \cos \theta
\end{align}$
We have the double angle formula $\sin 2\theta =2\sin \theta \cos \theta $. Thus, the expression is simplified as:
$2\sin \theta \cos \theta =\sin 2\theta $
Hence, the left side is equal to the right side.
$2\sin {}^{3}\theta \cos \theta +2\sin \theta {{\cos }^{3}}\theta =\sin 2\theta $
